NMC Sample Problems: Grade 11
1.
Which one of the following functions has different domain?
(a)
2.
1
100x
(b)
(c) x2 + (y − 12 )2 =
(e) x2 + (y − 12 )2 =
sin x
x(x2 +1)
(d)
(x+1)(x2 +2)
x3 +1
(e)
(x+1)(x3 +1)
x4
(b) (x + 12 )2 + y 2 =
5
4
√
5
2
5
4
(d) (x − 12 )2 + y 2 =
√
5
2
√
5
2
What is the solution set of the following equation: e10x+1 = ex+10 ?
(a) {0}
4.
(c)
Find the equation of the circle centered at ( 21 , 0) passing through the point (0, 1).
(a) (x − 12 )2 + y 2 =
3.
√4
x x2 +1
(b) {1}
(c) {1, −1}
(d) {1, 2}
(e) {0, 1}
Find the polar coordinates (r, θ) for the point with rectangular coordinates (x, y) = (−2, 0).
(b) (2, π2 )
(a) (−2, 0)
(c) (2, π)
(d) (2, 3π
2 )
(e) (2, 2π)
5.
√
Find the rectangular coordinates (x, y) for the point with polar coordinates (r, θ) = ( 2, 5π
4 ).
√
√
√ √
(a) (−1, −1)
(b) (1, 1)
(c) (− 2, − 2) (d) ( 2, 2)
(e) (2, 2)
6.
When cos θ =
(a)
7.
45
81
(b) − 45
(c)
3
5
(d) − 35
(e) ± 35
(b)
46
81
(c)
47
81
(d)
48
81
(e)
49
81
There are 1 red ball, 2 blue balls, and 4 white balls in a bag. If James takes out four balls at once,
then what is the probability that he takes out at most one white ball?
(a) 1
9.
and tan θ < 0, find cos(θ + π2 ).
If sin x + cos x = 13 , then what is the value of sin4 x + cos4 x?
(a)
8.
4
5
4
5
(b)
4
35
(c)
5
35
(d)
6
35
(e)
7
35
Let x and y be real numbers satisfying the following two conditions:
x−y =1
and x3 − y 3 = 16
Find x2 + y 2 .
(a) 9
(b) 10
(c) 11
1 of 8
(d) 12
(e) 13
NMC Sample Problems
10.
Grade 11
Let a8 , a7 , · · · , a1 and a0 be the coefficients of the expansion of the expression (x + 1)8 , that is
(x + 1)8 = a8 x8 + a7 x7 + · · · + a1 x + a0
Find the alternating sum a8 − a7 + · · · − a1 + a0 .
(a) −2
11.
510 −1
4×59
(b)
510 +1
4×59
(c)
410 +1
4×59
(d)
410 −1
4×59
(e)
510
4×59
(b) 4
(c) 5
(d) 6
(e) 0
(b) 3000
(c) 3400
(d) 3700
(e) 4000
Let a, b and c be three real roots of x3 + x2 − 3x + 1 = 0. Find p(a) + p(b) + p(c), where
p(x) = x4 + 4x3 − 7x + 4.
(a) −2
15.
(e) 2
How many seven digit numbers contain the digit pattern “2016” exactly once?
(a) 2700
14.
(d) 1
What is the remainder when 1! + 2! + 3! + · · · + 2016! is divided by 7?
(a) 3
13.
(c) 0
Find the sum 1 + 5−1 + 5−2 + · · · + 5−9 .
(a)
12.
(b) −1
(b) −1
(c) 0
(d) 1
(e) 2
Suppose that in the following triangle the length of each of the sides x, y is a positive integer. When
the ratio x : y is 5 : 6, find the smallest integer area of the triangle. (Note: Figure not drawn to
scale! )
x
30◦
y
(a) 10
16.
(c) 30
(d) 40
(e) 50
p
√
Suppose a, b, c and d are integers, and x = − 2 − 3 is a solution of the equation x4 + ax3 +
bx2 + cx + d = 0. What is the sum a + b + c + d?
(a) −3
17.
(b) 20
(b) 6
(c) −9
(d) 12
(e) −15
What is the remainder when x2016 + 1 is divided by x − 1?
(a) x2015 + 1
(b) x − 1
(c) 1
2 of 8
(d) 2
(e) 3
NMC Sample Problems
18.
19.
Find the equation of the straight line which passes through the point (20, 16) and is perpendicular
to 5x + 4y + 1 = 0.
(a) 5x + 4y − 164 = 0
(b) 5x − 4y − 36 = 0
(d) 4x − 5y = 0
(e) 4x + 5y = 0
22.
2017
4032
2016
8032
(c)
2017
8032
(d)
8032
2017
(d) 6 (six solutions)
(e) ∞ (infinitely many solutions)
(e) (3, 2)
(e)
2016
4032
(c) 4 (four solutions)
A ball is thrown vertically upward and its height after t seconds is h(t) = 64t − 32t2 feet. Find the
maximum height reached by the ball.
(b) 32 feet
(c) 64 feet
(d) 128 feet
(e) 150 feet
x+1
The function f (x) = 4x−3
has one horizontal asymptote, say y = a, and one vertical asymptote,
say x = b. Find a + b.
(b) 0
(c) 1
(d) 3
(e) 5
4
5
(e) 1
If 10x = 5, then which of the following is 10−2x+1 ?
1
5
(b)
2
5
(c)
3
5
(d)
If sec x = 5 and 0◦ < x < 180◦ , then what is sin x?
(a)
26.
(b)
(b) 2 (two solution)
(a)
25.
(d) (−2, −3)
(a) 0 (no solutions)
(a) −1
24.
(c) (−2, 3)
Which one is NOT a possible number of the real solutions of the equation ax2 + b − sin x = 0 (a, b
are real numbers and a 6= 0.)?
(a) 0 feet
23.
(b) (2, −3)
Find the value of the product:
1
1
1
1
1− 2
1 − 2 ··· 1 −
1− 2
2
3
4
20162
(a)
21.
(c) 4x + 5y − 160 = 0
Find the vertex point of the parabola y = x2 − 4x + 1.
(a) (2, 3)
20.
Grade 11
√
2 6
5
√
(b) − 2 5 6
(c)
1
5
(d) − 15
(e)
√
2 2
5
If sin x − cos x = 12 , then what is the value of sin3 x cos x − cos3 x sin x?
(a)
√
3 3
4
√
(b) − 3 4 3
√
(c) ± 3327
3 of 8
(d)
3
8
(e) − 38
NMC Sample Problems
27.
Evaluate log1/16 8?
(a) −2
28.
(b) − 12
(b) 1108
(c) 1208
(b) 7
Evaluate the following:
1+i
√
2
2
5
(b) 1
(c)
(b) 11
√
2(1−i)
2
(c) 10 +
√
(d)
2
2(1+i)
2
(d) 5
√
(e) −
2(1+i)
2
(e) 10 −
(b) 420
(c) 210
√
2
(b) {−1, −2}
x−2
(d) 1
(e) 0
(d) {− 32 }
(e) {} (No solution)
= 1.
(c) { 23 }
(b)
3
10
(c)
1
5
(d)
4
5
(e) 0
Find the sum of all integers between 1 to 100 (inclusive) that are not multiples of 4.
(a) 3747
36.
(e) 15
A committee composed of Alice, Mark, Ben, Connie and Francisco is about to select three representatives randomly. What is the probability that Alice and Mark are included in the selection?
(a)
35.
(e) 0
2016
Find the solution set of the equation 2x−1
(a) {1, 2}
34.
3
4
In the expansion of (x + y)10 , what is the coefficient of x4 y 6 ?
(a) 960
33.
(e)
(d) 13
Find the maximum value of sin x − cos x + 10.
(a) 10
32.
1
2
(d) 1308
(c) 10
√
(a) 0
31.
(d)
Find the sum of the maximum and minimum values of the expression 4 cos(2x + 1) + 5.
(a) 3
30.
(c) − 34
Find the remainder when 1 + 2 + 3 + · · · + 2015 + 2016 is divided by 2016.
(a) 1008
29.
Grade 11
(b) 3748
(c) 3749
(d) 3750
(e) 3751
Evaluate the following:
2
3
4
2016
log2016
+ log2016
+ log2016
+ · · · + log2016
1
2
3
2015
(a) −10
(b) −5
(c) 1
4 of 8
(d) 5
(e) 10
NMC Sample Problems
37.
How many functions which are NOT one-to-one are there from the set X = {a, b, c} to the set
Y = {1, 2, 3, 4}?
(a) 10
38.
Grade 11
(b) 24
(c) 30
√
13
3
(c)
(b) 21
The rational function f (x) =
(a) 4
41.
(b)
√
1+ 13
3
(d)
√
2+ 13
3
(e) 1
What is the median of the numbers in the set {2, 6, 10, 14, · · · , 38, 42} (The set consists of the first
few terms of the arithmetic progression dn = −2 + 4n, n = 1, 2, ... )? (For example, the median of
the numbers 1, 2, 3 is 2 and the median of the numbers 1, 2, 3, 4 is 2.5)
(a) 22
40.
(e) 64
Let z = a + 2i be a complex number, where a is a positive real number(i2 = −1). If the imaginary
part of z 2 and z 3 are equal, then what is a?
(a) 0
39.
(d) 40
(c) 20
px+q
rx+s
(b) −4
(d) 19
has the inverse function f −1 (x) =
(c) 0
(e) 18
x+4
2x+1 .
(d) 5
Find p + q + r + s.
(e) −5
Find sin B of the following triangle. (Note: Figure not drawn to scale! )
A
3
2
B
(a)
42.
7
16
11
16
√
(c)
15
16
(d)
√
2 15
16
(e)
√
3 15
16
An employee of a computer store is paid a base salary $1,700 a month plus a 2% commission on all
sales over $4,200 during the month (For example, if the gross sales during the month is $5200, then
the commission is $20). How much must the employee sell in a month to earn a total of $2,000 for
the month?
(a) $16,200
43.
(b)
C
4
(b) $17,200
(c) $18,200
(d) $19,200
(e) $20,200
A speedboat takes a half hour longer to go 10 miles up a river than to return. If the boat cruises
at 20 miles per hour in still water, what is the rate of the current? (The units are miles/hour.)
√
√
√
(a) 10( 2 − 1)
(b) 20 2
(c) 20( 2 − 1)
(d) 6
(e) 8
5 of 8
NMC Sample Problems
44.
Grade 11
The domain of the function
f (x) = log2016 (log2016 (log2016 (log2016 x)))
is {x|x > c}. What is the value of c?
(a) 0
45.
(d) 5
(e) 0
(b) 820
(c) 630
(d) 440
(e) 250
(b) −10
√
3 is a solution of the equation x3 + mx2 + nx + 12 = 0.
(c) 5
(d) −5
(e) 0
(b) 14
(c) 15
(d) 16
(e) 17
(b) 77
(c) 101
(d) 110
(e) 111
A polynomial f (x) = x3 + ax2 + bx + c satisfies f (1) = 2, f (2) = 3 and f (3) = 4. What is the
remainder when f (x) is divided by x + 2?
(a) −63
51.
(c) 3
Suppose that f (x) is a polynomial with integer coefficients, having 2015 and 2017 as zeros. Which
of the following could possibly be the value of f (2022)?
(a) 35
50.
2016
(e) 20162016
Harry performed the calculation: 7 × 7 = 34. It turns out Harry’s calculation is correct in base b
(b > 7). Find the value b.
(a) 13
49.
(b) 2
Suppose that m, n are integers and 3 +
Find m + n.
(a) 10
48.
(d) 20162016
Which one of the following is the greatest?
(a) 1010
47.
(c) 2016
Find the sum of all solutions of the equation |x − 1| + |x − 2| = 2.
(a) 1
46.
(b) 1
(b) −62
(c) −61
20164 + 3 · 20163 − 3 · 20162 + 6 · 2016 + 8
.
20163 − 20162 + 2016 + 2
(a) 2016
(b) 2017
(c) 2018
(d) −60
(e) −59
(d) 2019
(e) 2020
Evaluate
52.
Find the remainder when 20162016 is divided by 7.
53.
If log4 9 = a, then what is log27 2 in terms of a?
54.
Let a and b be integers which satisfy
p
√
√
19 + 6 2 = a + b 2. Find a + b.
6 of 8
NMC Sample Problems
55.
Grade 11
Let f be a function satisfying the following equation:
f (1 + x) + xf (1 − x) =
1
for all real numbers x 6= 0.
x
Find f (4).
56.
Find all ordered pairs (x, y) of real numbers satisfying the following equations:
x2016 + 2y 2016 = x2015 + 2y 2015 = x2014 + 2y 2014 .
7 of 8
B KEYS
C
[1] (d)
[15] (c)
[29] (c)
[43] (c)
[2] (a)
[16] (a)
[30] (b)
[44] (d)
[3] (b)
[17] (d)
[31] (c)
[45] (c)
[4] (c)
[18] (d)
[32] (c)
[46] (d)
[5] (a)
[19] (b)
[33] (a)
[47] (b)
[6] (c)
[20] (a)
[34] (b)
[48] (c)
[7] (e)
[21] (e)
[35] (d)
[49] (a)
[8] (b)
[22] (b)
[36] (c)
[50] (c)
[9] (c)
[23] (c)
[37] (d)
[51] (e)
[10] (c)
[24] (b)
[38] (c)
[52] 0
[11] (a)
[25] (a)
[39] (a)
[53]
[12] (c)
[26] (c)
[40] (a)
[54] 4
[13] (d)
[27] (c)
[41] (e)
[55]
[14] (e)
[28] (a)
[42] (d)
[56] (0, 0), (0, 1), (1, 0), (1, 1)
1
3a
2
15